• NCERT Chapter
Question 9

# Let f : N → N be defined by State whether the function f is bijective. Justify your answer.

fN → N is defined as It can be observed that:  ∴ f is not one-one.

Consider a natural number (n) in co-domain N.

Case I: n is odd

n = 2r + 1 for some r ∈ N. Then, there exists 4r + 1∈N such that .

Case II: n is even

n = 2r for some r ∈ N. Then,there exists 4r ∈N such that .

∴ f is onto.

Hence, f is not a bijective function

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Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by . Is f one-one and onto? Justify your answer.

• Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
• Q:-

Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

• Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
• Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
• Q:-

Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

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Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by . Is f one-one and onto? Justify your answer.

• Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
• Q:-

Check the injectivity and surjectivity of the following functions:

(i) f : N → N given by f(x) = x2

(ii) f : Z → Z given by f(x) = x2

(iii) f : R → R given by f(x) = x2

(iv) f : N → N given by f(x) = x3

(v) f : Z → Z given by f(x) = x

• Q:- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
• Q:-

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

• Q:-

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?